Can I change the default increment amount for an object's Transform properties?
1) I haven't found any way to change the increment factor when clicking the arrows in the transform panel. I use metric units and at every click it snaps 1 cm (for Location), it is 0.01 if you don't use any unit type.

Personally if I want to move , rotate or scale an object with smaller values, for example if I want to move it 0.5 mm on the X axis I type in the values: "G"(for grab) -> "X" (on X axis) -> ".0005"(for 0.5 mm) -> RETURN

2) Snapping. 

By default snapping is set to "INCREMENT" and it means exactly an unit seen  in the 3D viewport. It doesn't have to be 1 Unit (Meter). 
These snapping values are set by the grid seen when you are in ortographic view (NUMPAD 5 to toggle):

 

If you zoom in , you'll see the grid changing from 1 unit to 0.1 units and then to 0.01 units and so on as you zoom in and also the Indication in the upper Left corner will change from Meters to Centimeters, Millimeters. Now if you move an object holding CTRL the snap will be relative to those units.

To fine even more your snap you can hold CTRL + SHIFT while transforming and this will make the snap be 10x shorter.

P.S. You can change the type of element to snap to from increment to vertex, edge, face volume using the shortcut CTRL + SHIFT + TAB